How do I recreate this shape?
Edit: The 'mesh mode view' is actually viewport wireframe view and does not display all the edges and vertices you should be able to see in Edit mode. I would suggest opening the attached blend file to take a closer look. 

I've made this visual guide which I hope helps more than the explanation you provided in your post. The mesh is displayed in three states; edit mode, with the subdivision modifier visible and without. 

(Since I'm not using the Math meshing addon, I'm using an ico sphere with an extra step to generate the required shape).

I've also attached the blend file, so you can see how it's set up precisely.

As a general guideline, you'll likely want to use insets only. Smoothing in edit mode isn't required since the Subdivision Surface modifier will display all edges as smoothed, unless you either crease them or add edge loops close by. Both options will sharpen your edges.

Using insets also means that your mesh topology likely will be (almost) quads only. This makes it easier to later edit mesh features, like changing the scale of edge loops or moving things around. You can make complex shapes with subdivision surface modifiers, while still having a relatively simple mesh.

As for scaling the inset faces, you're generally going to use Median Point and Individual Origins. Controlling their depth can be done using median point and scaling; the size of the faces themself perpendicular to their normals uses scaling along individual origin.

My resulting shape: